Optimal Design and -Concavity. Christian Ewerhart. April 15, *) University of Zurich; postal address: Chair for Information Economics

Transcription

1 Optimal Design and -Concavity Christian Ewerhart April 15, 009 *) University of Zurich; postal address: Chair for Information Economics and Contract Theory, Winterthurerstrasse 30, CH-8006 Zurich, Switzerland; phone: ; fax:

2 Optimal Design and -Concavity Abstract. Tools from advanced real analysis and the Prékopa-Borell Theorem are combined to derive a tight sufficient condition for regularity (R. Myerson, Optimal auction design, Mathematics of Operations Research 6, 1981, pp ). The conventional log-concavity condition arises as a special case. The approach allows various generalizations, for instance to multidimensional types. Regularity is verified explicitly for numerous new families of parameterized distributions. Economic applications are outlined, in particular to the robustness of the modified Vickrey auction. Keywords and Phrases. Virtual valuation, Myerson regularity, Generalized concavity, Prékopa-Borell Theorem, Mechanism design. JEL-Codes. D8 - Asymmetric and Private Information; D44 - Auctions; D86 - Economics of Contract: Theory; C16 - Specific Distributions.

3 1. Introduction Over more than three decades by now, the economic analysis of optimal design under asymmetric information, as suggested first by Mirrlees [5] in the context of income taxation, has been fruitfully applied to a broad range of important practical problems. Specific applications include, but are not limited to, retail customer screening (Mussa and Rosen [7]), regulatory policy (Baron and Myerson [8]), optimal reserve prices (Riley and Samuelson [3]), multi-unit auctions (Maskin and Riley [3]), negotiations (Bulow and Klemperer [11]), and bilateral trade (Myerson and Satterthwaite [9]). 1 In these papers, as well as in numerous others, the marginal contribution of a given type to the expected value of the objective function can be spelled out in a rather explicit way. Defined from a seller s perspective, for instance, this net contribution is given by the marginal revenue (cf. Bulow and Roberts [1]) or virtual valuation ( ) = 1 ( ), (1) ( ) where and, respectively, denote the probability density and cumulative distribution functions of a random variable that proxies the buyer s unknown willingness to pay. In many specific models, it is of value to be able to ensure that virtual valuations are monotone in the type. This is because in such a regular design problem (cf. Myerson [8]), the application of Pontryagin s maximum principle can be circumvented, and the optimal design may receive amoreintuitiveform. 1 Among the more recent applications of this technique are keyword auctions conducted by operators of internet search engines. See, e.g., Garg et al. [18]. An alternative motivation is the empirical analysis of bidding data. See Section 5.

4 A common way to ensure regularity of the designer s problem is to impose that the hazard rate ( ) (1 ( )) of the type distribution, i.e., the reciprocal of the ratio in equation (1), is monotone increasing in the interior of the support interval, or equivalently, that the reliability function 1 ( ) is logconcave. 3 This approach has two main advantages. On the one hand, there istheusefuleconomic factthatlog-concave density functions breed distributions that possess a monotone hazard rate (Bagnoli and Bergström [4], An [1]). This relationship allows identifying a large class of regular problems. On the other hand, the hazard rate has an immediate technical interpretation as a conditional density of failure, which relates the regularity problem to genuinely statistical issues (see, e.g., Barlow and Proschan [7]). However, the log-concavity condition, imposed on either the reliability function or the density, will sometimes be more restrictive than necessary. Specifically, if the decline in the density function that causes the hazard rate to fall over a certain interval is not too pronounced, virtual valuations may still be increasing in the type (cf. Maskin and Riley [3, Footnote 3]). Indeed, empirical research is sometimes based on specifications that, as will become clear later, have precisely this property (see, e.g., Laffontetal.[], or Baldwin et al. [5]). It might therefore be of value to find alternative criteria that improve upon log-concavity conditions. Intuitively, this will also strengthen numerous existing results obtained under the assumption of regularity. 3 A nonnegative function defined on R is log-concave if = { : ( ) 0} is a convex set and ln( ( )) is concave on. Log-concavity is equivalent to strong unimodality, and to being a Polyá frequency function of order. For definitions and references, see, e.g., An [1]. 3

5 Responding to this motivation, the present paper develops new criteria for regular design 4 that share the convenient properties of the log-concavity conditions, yet are significantly sharper. While special cases of these criteria can be proved directly using carefully chosen arguments, the most general results could be obtained through the following two-step approach. First, it is shown that virtual valuations are nondecreasing if and only if the reciprocal reliability function associated with the type distribution is convex. 5 This equivalence holds generally provided the underlying density function satisfies two very weak smoothness conditions. The point of this reformulation is that regularity is expressed in terms of a generalized concavity condition that is significantly weaker than the monotone hazard rate condition. Moreover, as a positive side effect, a statistical interpretation of regularity akin to the one known for the monotone hazard rate condition can be given. In a second step, a powerful method due to Caplin and Nalebuff ([13, 14]) is applied to reduce the condition on the distribution function to a sufficient condition on the density. Specifically, it is shown that a type distribution exhibits nondecreasing virtual valuations provided that the negative reciprocal square root transform of the underlying density function is concave. As a sufficient condition for nondecreasing virtual valuations, this square root criterion on the density function is considerably tighter than the corresponding log-concavity condition. Of course, the criterion on the density function can only be sufficient, 4 The related literature is discussed in Section 8. 5 For a twice differentiable distribution function, this result can be verified immediately by comparing the signs of (1 (1 ( ))) 00 and 0 ( ). 4

6 not necessary. A counterexample can be constructed that both verifies this point and allows an intuitive understanding of the remaining gap between the square root criterion and regularity. However, among sufficient conditions formulated in terms of generalized concavity of the density function, the square root criterion is the sharpest possible. Specifically, while the criterion demands that ( ) is concave for = 1, weaker conditions for any 1 do not generally imply monotone virtual valuations. Thus, in this sense, a tight condition for regularity has been identified. While the focus of the exposition will be on virtual valuations as defined through equation (1), the underlying theory extends naturally in numerous ways. Specific extensions concern richer environments for optimal design, strict monotonicity of virtual valuations, non-linear relationships between types and values, multi-dimensional types with externalities, harmonic mixtures, and density functions with jumps. The criterion on the density function is then applied to specific families of distributions for which the usual log-concavity condition has no bite. This leads to new distributions that allow regular design. For instance, the lognormal distribution, whose density is never log-concave, can be shown to be regular provided it is not too heavily skewed. Further examples include the Pareto distribution, Student s distribution, the Cauchy distribution, the F distribution, the beta prime distribution, the mirror-image Pareto distribution, the log-logistic distribution, the inverse gamma distribution, the inverse chi-squared distribution, as well as the Pearson distribution, where parameter constraints will be made explicit in all cases. Thus, the criterion allows going significantly beyond the range of elementary distributions that have 5

7 beenknowntoberegularbythevirtueoflog-concavityconditions. The results of the present paper imply that the assumption of regularity is significantly weaker than suggested by existing conditions. This will be illustrated using three example applications. Specifically, these will concern the optimality of the modified Vickrey auction, the sensitivity of optimal regulatory policy to reported information, and the existence of an optimal mechanism for bilateral trade. However, given the widespread use of the regularity assumption in the economic analysis of optimal design, the possible range of application is probably much larger. The rest of the paper is organized as follows. Section reviews the notion of generalized concavity and the Prékopa-Borell Theorem. Section 3 contains the central result of this paper, which relates the convexity of the reciprocal reliability function to the monotonicity of virtual valuations. The square root criterion on the density function is derived and discussed in Section 4. Section 5 deals with a number of extensions. Section 6 derives a useful reformulation of the square root criterion in terms of log-derivatives and applies the criterion to specific parameterized families of distributions. Representative applications are outlined in Section 7, while Section 8 discusses the related literature. Section 9 concludes.. Avriel concavity This section briefly reviews the notion of generalized concavity (Avriel [3]) and the Prékopa-Borell Theorem, both of which will be important technical instruments in the subsequent analysis. Among alternative notions of concavity (cf., e.g., Diewert et al. [16]), the specific definition stated below is 6

8 highlighted by the fact that concavity properties are passed on from densities to integrals. Caplin and Nalebuff [13, 14] were first in suggesting the use of generalized concavity in the economics literature..1. Elementary properties A real-valued function 0 defined on R is called -concave, forsomefinite 6= 0, if the set = { : ( ) 0} is convex, and the function is concave on. 6 The latter condition amounts to ( +(1 ) 0 ) ( ( ) +(1 ) ( 0 ) ) 1 () being satisfied for any 0 and for any (0; 1). The interior case =1captures the usual notion of concavity on. For limit cases =, =0,and =, the definition is extended by the respective requirement that be either uniform on,log-concave,orquasi-concave. 7 Higher values of the parameter [ ; ] correspond to more stringent variants of concavity (cf. Caplin and Nalebuff [13]). That is, a -concave function is also b -concave for all b, but not necessarily vice versa. In particular, concavity on is more stringent than log-concavity, which is turn is more stringent than -concavity for any 0. Forafunction twice continuously differentiable on, the condition of -concavity of, forsome finite R, isequivalentto ( ) 00 ( ) (1 )( 0 ( )) (3) for all. 6 See Rockafellar [33] for background information on convex and concave functions. 7 This is the definition used in the economics literature (cf. Caplin and Nalebuff [13]). Avriel s original notion can be retrieved through an exponential transformation. 7

9 Fix [ ; ]. For a nondecreasing -concave function 0 defined on R, and a concave real-valued function defined on R,thecomposite function 7 ( ( )) is -concave. 8 For a -concave function 0 defined on R, the function 7 ( ( )) is -concave for any affine transformation : R R. 9 Positive multiples 7 ( ) of a -concave function 0 are also -concave, where 0. It follows from these properties that any variable shift, rescaling, mirror-image, or truncation of a -concave probability density function on R is again -concave... The Prékopa-Borell Theorem Consider a real-valued function 0 integrable with respect to the Lebesgue measure on R. Then the univariate version of a theorem attributed to Prékopa [30] and Borell [10] says that if is -concave, for some [ 1; ], then the integral ( ) = Z (e ) e (4) is b -concave, where 1 for = b = (1 + ) for 1 for = 1. (5) The Prékopa-Borell Theorem contains as a special case the fact that logconcavity is passed on from a density function to the corresponding cumulative distribution function, which is an important fact with a wide range of economic applications (cf. Bagnoli and Bergström [4]). 8 For finite, this is essentially Theorem 5.1 in Rockafellar [33]. The extension to infinite is immediate. 9 This follows immediately from the definition of an affine transformation (cf., e.g., Rockafellar [33, Section 1]). 8

10 The multivariate version of the Prékopa-Borell Theorem is carefully stated and explained in Caplin and Nalebuff [13]. For the purpose of the present analysis, it will be sufficient to recall one particular implication of that theorem (cf. Dharmadhikari and Joag-dev [17, Theorem 3.1]). Specifically, marginal distributions of -concave multivariate distributions are b -concave for b = (1 + ) provided that 1, where 1 is the codimension of the boundary Characterizing monotone virtual valuations This section contains the central result of the present paper. Specifically, it will be shown that under mild restrictions concerning the smoothness of the density function, virtual valuations are monotone increasing if and only if the reciprocal reliability function of the underlying type distribution is convex. Taken for itself, this finding sharpens the monotone hazard rate condition that is sufficient for regularity, yet not necessary. Consider a random variable, thetype, that takes on values in a nonempty open interval R with probability one. 11 Associated with is the cumulative distribution function (or c.d.f.) defined via ( ) =prob[ ] on the support closure { } of in the extended real line R { }. 1 The reliability function associated with is defined on { } through =1. A real-valued function 0, integrable with respect to the Lebesgue measure on R, is called a probability density function (or p.d.f.) 10 An interesting application concerns convolutions. For instance, while the convolution of a log-concave function with an arbitrary function need not be log-concave (cf. Miravete [4]), the convolution of two log-concave functions is always log-concave. Similar properties hold for general -concave functions (see Dharmadhikari and Joag-dev [17], or Borell [10]). 11 Multi-dimensional types will be considered in Section 5. 1 Thus, for example, {(0; )} =[0; ], etc. 9

11 for if Z ( ) = ( ) (e ) e (6) for any,where ( ) = ( ; ]. A density function defined on is strictly positive, inshort 0, if ( ) 0 for all. For a random variable and a density function 0 for, a function defined on is called a virtual valuation associated with the pair ( ) provided that ( ) = ( ) ( ) on. In the sequel, when there is no danger of confusion, the reference to will be dropped. Two assumptions on the smoothness of the density function will be imposed. Both conditions will be trivially satisfied for any p.d.f. that is continuously differentiable on. However, as pointed out by An [1], there are situations in which differentiability may be a restrictive assumption. This is the case, for instance, in non-parametric modeling, in models with endogenous type distributions, or when mixing over different distribution supports. In these situations, it is of interest to have a theory that does not require a differentiable density function. Afunction definedonanopenset R will be called Zygmund right-continuous in if lim 0+ ( + ) exists and lim sup ( ) ( ) = lim ( + ), (7) where lim sup denotes the (potentially infinite) upper limit. The first assumption reads as follows. Condition (Z). is Zygmund right-continuous at any. This smoothness condition excludes the possibility that the density function has points that would interfere with the regularity assumption, but do not 10

12 matter for reliability. For instance, if a continuous density function is manipulated by lowering its value at a single point, then the virtual valuation will become non-monotone, even though the reliability function does not change. This type of inessential discontinuity is excluded by condition (Z). The second condition is harder to justify from first principles, but it seems to be indispensible. 13 Intuitively, the derivative of the density, broadly understood, needs to be sufficiently informative to identify the density up to a constant. For a function definedonanopenset R, write + ( + ) ( ) ( ) = lim sup [ ; ] (8) 0+ for the right-hand upper Dini derivative of at a point. Condition (CL). Thereisacountableset such that + ( ) for all \. From the following result, everything else said later follows more or less immediately. Proposition 1. Consider an arbitrary random variable that takes on values in an open interval R with probability one, and let 0 be a density function for. Then ( ) 0 on. Moreover, provided that both (Z) and (CL) are satisfied, is nondecreasing if and only if 1 is convex. Proof. To show the first assertion, assume to the contrary that ( ) =1for some. Since is nondecreasing, (b ) =1for any b such that b. Thus, would be constant on a nondegenerate subinterval of, in contradiction to the assumption that 0. Thus, ( ) 0 for all. 13 In technical terms, condition (CL) excludes the so-called Cantor-Lebesgue anomaly, also known as the devil s staircase, that appears in the theory of integration (cf., e.g., Saks [35, p. 05]). 11

13 Only if. Assume that is nondecreasing. Then the right-hand upper Dini derivative satisfies + + ( + ) ( ) ( ) = lim sup 0+ 0 on. Moreover, from ( ) = (1 ( )) ( ), ½ ( + ) ( ) = 1 + lim sup 0+ ( + ) 1 = 1 + lim sup ( ) ( ) + 1 ( ) ¾ ( ) ½ ( + ) ( ) + ( + ) ( + ) ( ) (9) (10) ¾. (11) From the integral representation (6), it follows that the derivative of is well-defined a.e. in, and that when it exists, 0 = (cf., e.g., Royden [34, Ch.5,Thm.10]).Moreover, 0 is right-continuous by (Z). Hence, + ( ) = a.e. + (1 ( )) + ( ) ( ), (1) where = a.e. stands for equality for almost every. Next, define a function on via ( ) ( ) = (13) (1 ( )) for. The right-hand upper Dini derivative of can be written as ½ ¾ + 1 ( + ) ( ) = lim sup 0+ (1 ( + )) ( ) (14) (1 ( )) ½ 1 ( + ) ( ) = lim sup + 0+ (1 ( + )) ¾ ( ) (1 ( )) (1 ( + )) +. (15) (1 ( )) Recall that is continuous (cf., e.g., Royden [34, Ch. 5, Lemma 7]). More- 1

14 over, ( ) 1 for. Hence, ½ + 1 ( + ) ( ) ( ) = lim sup + (1 ( )) 0+ ¾ ( ) (1 ( )) (1 ( + )) +. (16) (1 ( )) Since 0 ( ) = a.e. ( ), + ( ) = a.e. + ( )(1 ( )) + ( ) (1 ( )) 3. (17) Comparing (1) and (17) yields that + ( ) 0 if and only if + ( ) 0 for almost any. In particular, + ( ) 0 a.e. in. Furthermore, exploiting condition (Z) on and the continuity of 1, it follows from (13) that lim sup ( ) ( ) lim sup ( + ). (18) at any. Finally, condition (CL) holds for. Indeed, this follows from condition (CL) on and from (16) by noting that is nondecreasing. By the discussion following Theorem 7.3 in Saks [35, Chapter VI], a function satisfying + ( ) 0 a.e. in, inequalities (18) at any, aswellas condition (CL), must be nondecreasing. Hence, is monotone increasing. By the chain rule, (1 ( )) 0 = a.e. ( ). Fix some nondegenerate compact interval [ 0 ; 00 ]. By Royden [34, Ch. 5, Thm. 14 and Ex. 14.c], the function 1 is absolutely continuous on [ 0 ; 00 ]. Hence, for [ 0 ; 00 ], Z 1 1 ( ) = 1 1 ( 0 ) + (e ) e. (19) 0 An integral over a nondecreasing function is convex (cf. the proof of Rockafellar [33, Thm. 4.4]). Thus, 1 is convex on any compact subinterval of, hence convex on the interval. 13

15 If. Conversely, assume that 1 is convex on. By Royden [34, Ch. 5, Prop. 17], the left derivative of 1 is well-defined and nondecreasing in, and equals the derivative of 1 except at a countable set. Almost everywhere in, thederivativeof1 is given by. Thus, there is a null set 0 such that is nondecreasing on \ 0. It is claimed that + ( ) 0 for any \ 0. Fix \ 0. Since 0 is a null set, any open neighborhood of contains an element 0 \ 0 such that 0. Hence, there is a decreasing sequence { } =1 in \ 0 converging to. Clearly, + ( + ) ( ) ( ) ( ) ( ) = lim sup lim sup 0. (0) 0+ Thus, + ( ) 0 a.e. in. As shown above, this implies + ( ) 0 a.e. in. It is claimed now that lim sup ( ) ( ) lim sup ( + ) (1) at any. But this follows from condition (Z) for and from the continuity of. Also condition (CL) holds for, as follows from (11) by noting that is nondecreasing, and that is right-continuous. Thus, by another application of Saks [35, Ch. VI, Thm. 7.3], is nondecreasing. The convexity condition on the reciprocal reliability function 1 allows a statistical interpretation that resembles the one known for the monotone hazard rate. Specifically, the value of the reciprocal reliability function can be understood as the conditional probability density that a given machine, that has been functional up to time, isthe next to break down. In the differentiable case, the first-order change of the reciprocal reliability is the 14

16 zoom rate =. 14 Intuitively, this is the rate by which the conditional probability density of breaking down next increases over time. Using this terminology, a type distribution satisfying the assumptions of Proposition 1 is regular if and only if the associated zoom rate is monotone increasing. 4. A sufficient condition for Myerson s (1981) regularity It is often useful to have a criterion for regularity that can be applied directly to the density function. For instance, the density may be given in explicit form, while the cumulative distribution function is an intractable integral. To derive a sufficient condition on the density function, I apply the Prékopa- Borell Theorem (cf. Section ). Luckily, the derivation does not add any differentiability assumptions, and also the smoothness conditions discussed in the previous section need not be imposed again. Proposition. Consider a random variable with a cumulative distribution function that allows a density function 0. Then is nondecreasing if 1 is convex. Proof. Assume that is ( 1 )-concave. Let = denote the mirror image of. Then the cumulative distribution function of is given by ( ) 1 ( ). A density function for is given by ( ) = 0 ( ) = ( ). Clearly, also is ( 1 )-concave. By the Prékopa-Borell Theorem, is ( 1)- concave. Equivalently, 1 is convex in. Thus,1 in convex in. Since must be continuous in (cf. Rockafellar [33, Thm. 10.1]), condition (Z) holds. Moreover, the right derivative of the convex function 1 is finite in by Theorem 4.1 in Rockafellar [33]. This delivers (CL). The assertion 14 For example, the Pareto distribution with parameter =1has a constant zoom rate. 15

17 now follows from Proposition 1. The criterion implied by Proposition is quite convenient to apply in examples because, as will become clear in Section 6, it can be reformulated so that it makes use of the same derivatives as the log-concavity condition on the density function. Moreover, the square root criterion is significantly sharper than the log-concavity condition (cf. the Introduction). Of course, generalized concavity of the density function is not a necessary condition for increasing marginal valuations. As an example, consider the convex mixture ( ) =(1 )+ ( ) of a uniform distribution on [0; 1] and some bimodal distribution on [0; 1], given by a continuously differentiable density function. It is clear that for 0 small enough, virtual valuations associated with must still be increasing. However, provided that 0, the mixed density will be bimodal, and therefore not quasi-concave. Thus, for 0, the density function cannot be -concave for any [ ; ]. Evidently from this example, what generalized concavity demands too much of the distribution is that a strictly convex section of the density function with a small slope must have density values sufficiently small to pass the test. 15 However,thesquarerootcriterionisthetightestpossibleamongconditions formulated in terms of generalized concavity of the density function. Tovalidatethispoint,itsuffices to present, for any 1, a density function that is -concave, yet exhibits virtual valuations that are declining over a nondegenerate subinterval of. Such an example is the Pareto dis- 15 Another reason why generalized concavity of the density function is not necessary for regularity is continuity. See Section 5 for an extension to density functions with jumps. 16

18 tribution with parameter 0 1 (cf. Section 6), whose density function is -concave for = 1, yet has everywhere declining virtual valuations. 1+ Thus, no further tightening of the square root condition is feasible among the generalized concavity conditions on the density function Extensions 5.1. Virtual costs and weighted objectives In addition to the virtual valuation, Myerson and Satterthwaite [9] consider also the virtual cost ( ) = + ( ) () ( ) associated with a buyer of type, wheretheratio ( ) ( ) is known as the reversed hazard rate. Still another variant of the standard expression is obtained by Baron and Myerson [8], who maximize an objective function that assigns a positive weight to both welfare and the firm s profit (cf. Section 7). To accommodate these possibilities, let ( ) = ( ), (3) ( ) where R are general parameters. 17 Note that ( 1 1) ( ) and ( 1 0) ( ). Propositions 1 and extend in an essentially straightforward way. Proposition 3. Let 1. Provided that 0 satisfies (Z) and (CL), ( 1) is nondecreasing in if and only if is -concave, with =. 16 Similar arguments show that log-concavity of the density function, while not necessary for the monotone hazard rate condition, is the tightest criterion possible for it within generalized concavity conditions on the density function. 17 Needless to say, the term may be replaced by an arbitrary function of, provided its slope is uniformly bounded by. 17

19 Similarly, provided that the mirror-image of 0satisfies (Z) and (CL), ( 0) is nondecreasing in if and only if is -concave, with =. In particular, under this condition, is nondecreasing if and only if is ( 1)-concave. Finally, ( ) is nondecreasing in for any [0; 1] if is -concave, with = (1 + ). In particular, is nondecreasing if is ( 1)-concave. Proof. Assume first that is twice continuously differentiable on, and that = 0.Fix 1. Differentiating (3) yields that ( )( ) 0 if and only if (1 + ) ( ) +( ( )) 0 ( ) 0. (4) For = and =1, inequality (4) says that is -concave. Hence, is ( )-concave if and only if ( 1) is nondecreasing in. For the case =0, consider the mirror image of, with cumulative distribution function ( ) =1 ( ) and probability density function ( ) = ( ). Then(4) reads (1 + ) ( ) +(1 ( )) 0 ( ) 0. (5) Thus, is ( )-concave if and only if ( 0) is nondecreasing in. But clearly, is ( )-concave if and only if is ( )-concave. Hence, ( 0) is nondecreasing in if and only if is ( )-concave. From the equivalences shown so far, it follows that ( 1) + (1 ) ( 0) = ( ) ( ) (6) is nondecreasing for any [0; 1] if and only if both and are ( )- concave. By the Prékopa-Borell Theorem, sufficient for being ( )-concave 18

20 is that is -concave for = (1 + ), providedthat 1. By a straightforward extension of the argument used in the proof of Proposition, the same condition is sufficient for being ( )-concave. This proves the assertions of the proposition in the case that is twice continuously differentiable. The extension to or satisfying merely (Z) and (CL) is obtained by following the lines of the proofs of Propositions 1 and in this somewhat generalized setting. The details are omitted. Proposition 3 generalizes the well-known fact (cf. Bagnoli and Bergström [4]) that distributions possessing a log-concave p.d.f. feature a nondecreasing hazard rate as well as a nonincreasing reversed hazard rate. For =1, for instance, the result says that the square root criterion implies virtual valuations and virtual costs to be monotone increasing in the type. 5.. Strict monotonicity In some applications, it is necessary to ensure that virtual valuations are not only nondecreasing, but increasing in the type. To deal with this point, the following definition is useful. For [ ; ), call a function strongly -concave if is 0 -concave for some 0 ( ; ] such that Proposition 4. The assertions of Proposition 3 remain true if all occurrences of -concave, including those for specific values,arereplacedby strongly -concave, and simultaneously all occurrences of nondecreasing arereplacedby increasing. Proof. Assume that is strongly -concave for =. By definition, 1+ there exists a 0 such that is 0 -concave. Hence, is 00 -concave for any 18 This definition should not be confused with strong concavity (cf. Diewert et al. [16]). 19

21 00 0.Choosesomeb ( 1; ) close to. Clearly, b,andb 1+ approaches as b. Byshiftingb closer to, if necessary, is b -concave. By Proposition 3, ( b ) is nondecreasing in. The assertion follows now from ( ) = ( b )+( b ). Proposition 4 allows an immediate application to the empirical analysis of bidding data. Guerre et al. [19] show that bids in the first price auction can be rationalized as a Bayesian Nash equilibrium in the independent private value paradigm if and only if + 1 ( ) ( ) = ( 0) (7) is increasing in, where is the number of bidders, while and, respectively, denote the density and distribution functions associated with the distribution of bids. Thus, for any strongly -concave with =, or somewhat less tightly, for any strongly b -concave with b = (1 + ), there exists a (unique) underlying distribution of valuations such that and, respectively, characterize the equilibrium distribution of bids for bidders Nonlinear valuations In some papers, such as Bulow and Klemperer [11], the analysis leads to the more general notion of virtual valuation ( ) = ( ) 1 ( ) 0 ( ), (8) 0 ( ) where ( ) is the value of type. 19 Proposition 3 does not apply directly unless is linear in. However, provided that both 0 and 0 are differen- 19 In fact, both the value and the type distribution may condition on additional signals, which are dropped here to keep the notation at a minimum. 0

22 tiable, 0 ( ) = 0 ( ) (1 ( )) 00 ( ) 0 ( ) 0 ( ) 1 ( ) 00 ( ). (9) 0 ( ) Thus, a sufficient condition for monotonicity of is that be increasing and concave, and that 1 be convex. In particular, under these conditions, the square root criterion can be used to verify monotonicity of Multidimensional types with externalities The regularity condition has been employed also in settings with type distributions of dimension. For instance, Jehiel et al. [1] require that the symmetric design problem be regular, i.e., that be increasing in, where Z ( ) = ( + 1 X 1 ) (30) for a given multidimensional density function. To illustrate the criterion in this environment, note that Proposition 4 yields as a sufficient condition 6= for regularity that is strongly ( 1 )-concave. Given that the change of variable in the argument of effective in (30) is an affine transformation of R, regularity follows from the multivariate version of the Prékopa-Borell Theorem (cf. Section ), provided that is strongly -concave, where satisfies (1 + ( 1)) = 1.Thus,if a -concave for 1,then +1 the multidimensional problem will be regular Harmonic mixtures Linear convex mixtures of regular distributions need not be regular in general. For example, while exponential distributions are regular, the nondegenerate mixture of two exponential distributions with different parameters is not reg- 1

23 ular for sufficiently high types, as can be shown directly by differentiation. 0 However, a weaker property holds. For distribution functions and, define the harmonic mixture with respect to a given parameter (0; 1) by ( ) =1 ( ( ) +(1 ) ( )), with the straightforward interpretation if ( ) =0or ( ) =0. It is then clear that is a ( 1)-concave distribution function if and have this property. Moreover, a density function for satisfying (Z) and (CL) can be defined provided and allow densities with the same properties. Thus, the class of sufficiently smooth regular distributions is closed under harmonic mixtures of the associated reliability functions. In fact, the axioms of a mixture set (cf. Herstein and Milnor [0]) are satisfied Density functions with jumps Probably the best-known example for a decline in the virtual valuation is a piecewise continuous density function that jumps downwards at a given point. Upwards jumps, however, do not generally invalidate regularity. To better understand this asymmetry, consider two distribution functions and such that associated reliability functions and are reciprocally convex. Since the pointwise maximum of two convex functions is convex (cf. Rockafellar [33, Theorem 5.5]), max{1 1 } =1 is convex, where =max{ }. Moreover, it is readily checked that for densities and corresponding to and, and satisfying conditions (Z) and (CL), a density function for 0 This is an instance of a more general phenomenon. Indeed, the class of -concave functions is not closed under addition for any 1. In particular, families of distributions with monotone increasing hazard rate or log-concave density function, respectively, are not closed under convex mixtures. Positive results hold for decreasing hazard rate distributions and for distributions with log-convex density function. See Barlow et al. [6], Borell [10], andan[1].

24 exists, such that also satisfies (Z) and (CL). Thus, regular type distributions as considered in Proposition 1 are closed under the formation of pointwise maxima of the associated distribution functions. Moreover, at points where and cross, the graph of typically exhibits an upwards jump. For instance, if is a proper mean-preserving spread of,andboth and are regular as well as continuous, then this technique glues together thelefttailof with the right tail of at the interior intersection point b of and, causing the resulting regular density function to jump upwards at b from (b ) to (b ). 6. Parameterized families of distributions This section derives and documents regularity properties of probability distributions commonly employed in statistic and economic modeling. The main tool will be a differentiable version of the square root criterion that will be stated below A reformulation of the square root criterion In the differentiable case, the square root criterion can be expressed alternatively in terms of the first and second log-derivatives of the density function. Specifically, for a twice differentiable density function at a point such that 0 ( ) 6= 0, consider the concavity measure ( ) = ln ( ). (31) ln ( )) ( The range of reflects generalized concavity properties of in the following way. 1 For some of these distributions, concavity properties have been discussed before (cf., e.g., Caplin and Nalebuff [13]). However, the subsequent discussion covers broader parameter ranges and additional distributions. 3

25 Proposition 5. Let 0 be twice continuously differentiable on. Assume that 0 ( ) 6= 0on some open and dense subset 0. Then, for finite, the function is -concave in if and only if ( ) for all 0. Proof. Assume that ( ) for all 0. By straightforward calculation, ( ) = ( ) 00 ( ) 0 ( ) 1. Thus, ( ) 00 ( ) (1 )( 0 ( )) in 0. By continuity, this holds also in. Hence, is -concave. The steps of the argument can be reversed, which proves the assertion. Proposition 5 is useful in examples because it reduces the determination of the concavity parameter to a straightforward maximization problem. Moreover, it exploits just those log-derivatives that are anyhow calculated to check log-concavity. 6.. Distributions with ( 1 )-concave density functions Bagnoli and Bergström [4] provide an extremely valuable overview over families of parameterized distributions with log-concave density function. Logconcave densities underlie, without any parameter restriction, the uniform, normal, exponential, logistic, extreme value, Laplace, Maxwell, and Rayleigh distributions. With restrictions to parameters, this list extends to power function, Weibull, Gamma, Chi-Squared, Chi, and beta distributions. These distributions all exhibit a nondecreasing hazard rate as well as a nonincreasing reversed hazard rate. Consequently, both and are monotone increasing for these distributions. Proposition and 3 imply an expansion of the list of parameterized families of distributions allowing regular design. As will be shown further below, 4

26 regular design is feasible also for the log-normal distribution, the Pareto distribution, the log-logistic distribution, Student s distribution, the Cauchy distribution, the F distribution, the beta prime distribution, the mirror-image Pareto distribution, the inverse gamma distribution, the inverse chi-squared distribution, and even for the Pearson distribution family, where parameter constraints ensure that these distributions are not too much skewed, and what is often equivalent, have moderate tails. In all these examples, the usual requirement of log-concavity on the p.d.f. is not strong enough to imply regularity. Table I offers an overview Distributions without ( 1 )-concave density functions Whenthesquarerootcriteriondoesnotapply,itismorecomplicatedto determine the monotonicity properties of virtual valuations and virtual costs. One possibility is that the underlying density function is nondecreasing so that, as an immediate consequence of the definition, must be increasing. Similarly, if is nonincreasing, then must be decreasing. Another possibility involves using concave transformations of the underlying random variable. E.g., in the case of the log-normal, Chi-Squared, and F distributions, ( ) = ( ( )) for some concave, sothat inherits log-concavity from some other distribution function. Further, monotonicity of or can sometimes be excluded by considering limit behavior at the boundary of the support interval. A final possibility is that the hazard rate or the reversed hazard rate have an explicit form. In the few remaining cases, I used numerical simulations. Limited information is contained in the support interval. From a result due to Barlow and Proschan [7, page 79, Exercise 11], more recently extended by Block et al. [9], it would follow that for either bounded from above or unbounded from below (or both), 5

27 Table II describes several distributions that lack a ( 1 )-concave density function Remarks on specific distributions In this subsection, will always refer to the p.d.f., and always to the c.d.f. of the respective distribution. It will be checked whether and are monotone increasing on or nonmonotone. 3 Extensions such as the consideration of strict monotonicity are essentially straightforward and will be omitted for reasons of space. Unless explicit reference is made to parameters, assertions are made with the implicit understanding that they hold for any fixed choice of parameters. Moreover, the results of Sections 4 and 5 will be used without explicit mentioning. The Pareto distribution can be defined on [1; ) by ( ) = 1,where 0isaparameter. Obviously, is -concave for = 1. In particular, +1 thesquarerootcriterionissatisfied if and only if 1. Directcomputation confirms that is nondecreasing if and only if 1. Moreover, since is decreasing, is increasing for any 0. The p.d.f. of the log-normal distribution is given for 0 by ( ) = 1 exp{ (ln ) }, (3) where R and 0 are the parameters of the corresponding normal distribution. It is straightforward to check that ( ) = ln ln 1 (ln ln ), (33) cannot be decreasing everywhere on. By symmetry, for either unbounded from below or bounded from above (or both), cannot be decreasing everywhere on. However, these results can never be helpful to identify regular distributions. 3 A function will be called nonmonotone if it is not monotone increasing on. 6

28 where =exp( ) is the mode of the log-normal distribution. In particular, ln is concave for ln 1+ln and convex for ln 1+ln. To check generalized concavity, one notes ( ) 4, wheretheinequality becomes an equality for ln =ln +, chosen so as to maximize (33). Thus, is -concave with = 4, whichisanewfinding in the literature. In particular, the square root criterion is satisfied if and only if. In terms of mean and variance of the log-normal distribution, =ln(1+ ),sothatboth is monotone increasing provided that the coefficient of variation satisfies 1, where 718 denotes Euler s constant. 4 For parameters violating this condition, is typically nonmonotone, as numerical calculations suggest. Moreover, since ( ) (ln ), where denotes the log-concave c.d.f. of the corresponding normal distribution, is log-concave, so that is increasing for all parameter values. Student s t distribution with 0 degrees of freedom is given on R by ( ) = +1 Γ( ) Γ( where Γ denotes the Gamma function, as usual. From +1 + )(1 ), (34) ( ) = ( +1) , (35) the density function is -concave with = 1.Inparticular,for 1, +1 both and are monotone increasing, even though the underlying density function is log-concave only on the interval [ ;+ ]. For 0 1, numerical calculations indicate that and are nonmonotone. 4 This implies, in particular, that the assumption =0 05 in Laffont et al. [] ensures regularity. 7

29 The Cauchy distribution is a distribution for =1,withdensityfunction ( ) = 1 (1+ ) 1. The log-density ln is neither concave nor convex. However, ( ) =( 1) ( ) is bounded from above by 1, hence is ( 1)-concave. In particular, both and are monotone increasing. The F distribution with 1 0 and 0 degrees of freedom is defined on (0; ) through ( ) =( 1 ) 1 ( ) Γ( 1+ ) Γ( 1 )Γ( ( 1 ) ) 1+. (36) It is well-known that is not log-concave (cf. Bagnoli and Bergström [4]). The concavity index reads ( ) = + {1 ( + }, (37) ( ) 1 ) where = ( 1 ) 1 ( +). (38) Clearly, is the mode of the distribution if 1. To determine the range of,notethat does not possess an interior strict maximum. 5 Hence, considering the limits of for =0and,onefinds that is - concave with =min{ 1 + }. An examination of the sign of yields that = if 1 1, while = + if 1, which adds generality to Borell s [10] finding. Thus, by the square root criterion, is monotone increasing for 1. On the other hand, for 1, asymptotically 0 ( ) as 0, sothat is nonmonotone in this case. As numerical computations show, may be nonmonotone also if. Furthermore, 5 Indeed, for 0, = 0, and 0, respectively, is strict monotone, constant, and a negative quadratic hyperbola. 8

30 using the same argument as in the case of the log-normal distribution, is always increasing. The density function of the beta prime distribution is given on R + by ( ) = ( ) 1 1 (1 + ),where 0 are parameters and denotes the beta function. One can check that ( ) = ( ), where is the c.d.f. of the distribution with parameters 1 = and =. Thus, using the finding derived above, is not log-concave, but -concave with = 1 1 increasing, whereas if 1 and with = 1 1+ if 1. Moreover, is is monotone increasing for 1, and otherwise nonmonotone, where the case 1 relies on a numerical calculation. For a parameter 0, the p.d.f. of the mirror-image Pareto distribution reads ( ) = ( ) 1,where 1. Clearly, if denotes the density function of the corresponding Pareto distribution, ( ) ( ) and ( ) ( ). Thus, is monotone increasing for any 0. Moreover, is monotone increasing if and only if 1. For any parameter 0, thepower function distribution on (0; 1) is given by ( ) = 1. Clearly, is -concave for =1 ( 1). Thus, both and are monotone increasing for 1. Infact,sincefor 1, the density function is decreasing, is monotone increasing for any 0. Differentiating shows that is nonmonotone for 1. The Weibull distribution is given by ( ) = 1 exp( ) for 0, where 0 is a parameter. Since is log-concave for 1, both and are monotone increasing for 1. Infact, is decreasing for 1, so is 9

31 nondecreasing for any 0. A straightforward examination of (1 )(1 + ) ( ) = (39) ( 1 exp( )) shows that is merely quasi-concave for 1. Indeed, is nonmonotone for 1, as a short calculation shows. For a parameter 0, thegamma distribution is defined on (0; ) by the density function ( ) = 1 exp( ) Γ( ). The associated concavity function reads ( ) =(1 ) (1 + ). Clearly, is negative for 1, so that and are monotone increasing under this condition. In fact, since is decreasing for 1, virtual costs are monotone increasing for any 0. On the other hand, for 1, asymptotically 0 ( ) as 0, sothat is nonmonotone. Indeed, for 1, the density function is only -concave with = 1. 1 The Chi-Squared distribution with parameter 0 is given on (0; ) by ( ) = 1 exp( ). (40) Γ( ) The distribution results from the Gamma distribution with parameter by a scale transformation that does not affect generalized concavity. 6 Hence, and are monotone increasing for, and nonmonotone for. Moreover, is monotone increasing for any 0. For a parameter 0, thechi distribution is specified via ( ) = 1 exp( ) (41) ( ) Γ( ) for 0. From ( ) = 1 (1 + ), (4) 6 Alternatively ( ) = ( ), where is the distribution function of the Chi distribution with parameter. 30

32 the density function is log-concave for 1, and -concave with = 1 (1 ) for 1. Thus, is monotone increasing for 1. For 1, asymptotically 0 ( ) as 0. Thus, is nonmonotone for 1. Moreover, using that is decreasing for 1, virtual costs are monotone increasing for any 0. The density function ( ) = 1 (1+ ) characterizes the log-logistic distribution for 0 and parameter 0. This density is not log-concave. In fact, the hazard rate ( ) ( ) = 1 (1 + ) is decreasing if 1, and first increasing, then decreasing if 1. However, from closed-form representations 1 ( ) =1+ and 1 ( ) =1+,itfollowsusing Propositions 1 and 3 that is monotone increasing if and only if 1, while is monotone increasing for any 0. To derive the concavity of the density function, one can check that the derivative of ( ) = (1 + )( +1) ( ( +1) +1) (43) does not vanish unless =1, so that the suprema of must be at the boundary. This yields that is -concave with = 1 if 1 and with +1 = 1 if 1. In particular, the square root criterion holds if and only if 1 1. For a parameter 0, theinverse gamma distribution is given on R + by ( ) =Γ( ) 1 1 exp( 1 ). The density is not log-concave. However, writing =1 (1 + ) for the distribution mode, the concavity function reads ( ) = (1 ( ) ). (44) ( ) Since is bounded from above by, the density is -concave for = 31

33 1. In particular, both 1+ and are monotone increasing for 1. In fact, since is log-concave for, and decreasing for,virtual costs are decreasing for any 0. Virtual valuations, however, are nonmonotone for 1, as numerical calculations suggest. The inverse chi-squared distribution has the density function ( ) = exp( 1 ( )) Γ( ) 1 (45) on R +,where 0 is a parameter. Clearly, ( ) = ( ), where denotes the density of the inverse gamma distribution with parameter =. In particular, the log-density ln is not concave for large values of, yet is -concave with =. Moreover, + is monotone increasing. is monotone increasing for, and numerically nonmonotone for. For parameters 0, 0, thebeta distribution on the interval (0; 1) is specified via Γ( + ) ( ) = Γ( )Γ( ) 1 (1 ) 1. (46) A straightforward calculation shows ( ) = (1 )(1 ) +(1 ) ((1 )(1 ) (1 ) ). (47) If 6= 1and 6= 1, 7 then has no interior maximum on (0; 1), ascanbe checked by differentiation, and so is -concave with = max ( ) =min{ 1 [0;1] 1 1 }, (48) 1 7 If = =1, then the distribution is the uniform distribution, which has a concave density function. If =1and 6= 1,then ( ) =1 (1 ), so the distribution is -concave with =1 ( 1). Similarly, if 6= 1and =1, then the distribution is -concave with =1 ( 1). 3

34 again a result not to be found in the literature. In particular, for 1 1 or 1 1, onefinds =(max{ ; } 1) 1, while for 1 1 or 1 1, the concavity parameter reads =(min{ ; } 1) 1 0. In sum, satisfies the square root criterion if and only if both 1 1 hold. If 1, asymptotically 0 ( ) as 0. Similarly, if 1, asymptotically 0 ( ) as 1. Thus, is nonmonotone if 1 or 1. Bysymmetry,also is nonmonotone in this case. The Arc-Sine distribution is a special case of the beta distribution with = =1. The density function on (0; 1) reads ( ) = 1 1 (1 ) 1. From (48), is only ( )-concave. Indeed, both and are nonmonotone, as shown more generally above. A density function belongs to the family of Pearson distributions if it is a solution of the ordinary differential equation 0 ( ) = ( )( ) ( ), where R is a parameter and ( ) = is a quadratic term with additional parameters 0 1 R. It is straightforward to check that ( ) = ( ) ( ). Since special cases have already been dealt with above, I confine myself to solutions with unbounded support, and such that is the mode of the distribution, i.e., ( ) 0. Under these conditions, is ( )-concave. By the square root criterion, both and are monotone increasing provided that 1.For 1, however, both and may be non-monotone. This follows from numerical findings obtained above for Student s distribution with 1, noting that this is a Pearson distribution of type VII with 1. 33

35 7. Applications Thepurposeofthissectionistoillustratesomeoftheeconomicimplications of the theory. Specifically, I will review models of optimal auction design, regulation under asymmetric information, and bilateral trade. For ease of reference, the notation in this section will follow the original papers Optimal auctions In Myerson [8], a single object is offered to potential buyers, where buyers valuations 1 of the object are distributed independently, according to density functions 1. Here, the virtual valuation of type reads ( )= ( ) 1 ( ) ( ) 0, (49) where 0 denotes the seller s reservation value, the c.d.f. of the underlying type distribution, and the revision effect function. In the symmetric case, i.e., if all bidders have an identical distribution of types, and with zero revision effect, the revenue-maximizing mechanism is a modified Vickrey auction, provided that is increasing in. If the problem is not regular, then the optimal mechanism is more difficult to interpret. It is known that type distributions with log-concave density function are regular. It follows from Proposition that any density function that is strongly ( 1 )-concave has this property. 7.. Regulation Baron and Myerson [8] solve the problem of regulating a monopolist with unknown cost parameter. For instance, might measure the firm s level of marginal costs. The regulator maximizes the sum of expected consumer surplus and a share [0; 1] of the expected profitforthefirm. The marginal 34